Calculus Examples

$lim_{x \to \infty} \frac{x^{4}}{3 x^{2} - 7 x}$

Step 1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 1.1

Take the limit of the numerator and the limit of the denominator.

$\frac{lim_{x \to \infty} x^{4}}{lim_{x \to \infty} 3 x^{2} - 7 x}$

Step 1.2

The limit at infinity of a polynomial whose leading coefficient is positive is infinity.

$\frac{\infty}{lim_{x \to \infty} 3 x^{2} - 7 x}$

Step 1.3

The limit at infinity of a polynomial whose leading coefficient is positive is infinity.

$\frac{\infty}{\infty}$

Step 1.4

Infinity divided by infinity is undefined.

Undefined

$\frac{\infty}{\infty}$

Step 2

Since $\frac{\infty}{\infty}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to \infty} \frac{x^{4}}{3 x^{2} - 7 x} = lim_{x \to \infty} \frac{\frac{d}{d x} [x^{4}]}{\frac{d}{d x} [3 x^{2} - 7 x]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

$lim_{x \to \infty} \frac{\frac{d}{d x} [x^{4}]}{\frac{d}{d x} [3 x^{2} - 7 x]}$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$lim_{x \to \infty} \frac{4 x^{3}}{\frac{d}{d x} [3 x^{2} - 7 x]}$

Step 3.3

By the Sum Rule, the derivative of $3 x^{2} - 7 x$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 7 x]$ .

$lim_{x \to \infty} \frac{4 x^{3}}{\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 7 x]}$

Step 3.4

Evaluate $\frac{d}{d x} [3 x^{2}]$ .

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Step 3.4.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

$lim_{x \to \infty} \frac{4 x^{3}}{3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 7 x]}$

Step 3.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$lim_{x \to \infty} \frac{4 x^{3}}{3 (2 x) + \frac{d}{d x} [- 7 x]}$

Step 3.4.3

Multiply $2$ by $3$ .

$lim_{x \to \infty} \frac{4 x^{3}}{6 x + \frac{d}{d x} [- 7 x]}$

Step 3.5

Evaluate $\frac{d}{d x} [- 7 x]$ .

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Step 3.5.1

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7 x$ with respect to $x$ is $- 7 \frac{d}{d x} [x]$ .

$lim_{x \to \infty} \frac{4 x^{3}}{6 x - 7 \frac{d}{d x} [x]}$

Step 3.5.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to \infty} \frac{4 x^{3}}{6 x - 7 \cdot 1}$

Step 3.5.3

Multiply $- 7$ by $1$ .

$lim_{x \to \infty} \frac{4 x^{3}}{6 x - 7}$

Step 4

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $x$ .

$lim_{x \to \infty} \frac{\frac{4 x^{3}}{x}}{\frac{6 x}{x} + \frac{- 7}{x}}$

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $x^{3}$ and $x$ .

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Step 5.1.1

Factor $x$ out of $4 x^{3}$ .

$lim_{x \to \infty} \frac{\frac{x (4 x^{2})}{x}}{\frac{6 x}{x} + \frac{- 7}{x}}$

Step 5.1.2

Cancel the common factors.

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Step 5.1.2.1

Raise $x$ to the power of $1$ .

$lim_{x \to \infty} \frac{\frac{x (4 x^{2})}{x^{1}}}{\frac{6 x}{x} + \frac{- 7}{x}}$

Step 5.1.2.2

Factor $x$ out of $x^{1}$ .

$lim_{x \to \infty} \frac{\frac{x (4 x^{2})}{x \cdot 1}}{\frac{6 x}{x} + \frac{- 7}{x}}$

Step 5.1.2.3

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{x (4 x^{2})}{x \cdot 1}}{\frac{6 x}{x} + \frac{- 7}{x}}$

Step 5.1.2.4

Rewrite the expression.

$lim_{x \to \infty} \frac{\frac{4 x^{2}}{1}}{\frac{6 x}{x} + \frac{- 7}{x}}$

Step 5.1.2.5

Divide $4 x^{2}$ by $1$ .

$lim_{x \to \infty} \frac{4 x^{2}}{\frac{6 x}{x} + \frac{- 7}{x}}$

Step 5.2

Simplify each term.

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Step 5.2.1

Cancel the common factor of $x$ .

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Step 5.2.1.1

Cancel the common factor.

$lim_{x \to \infty} \frac{4 x^{2}}{\frac{6 x}{x} + \frac{- 7}{x}}$

Step 5.2.1.2

Divide $6$ by $1$ .

$lim_{x \to \infty} \frac{4 x^{2}}{6 + \frac{- 7}{x}}$

Step 5.2.2

Move the negative in front of the fraction.

$lim_{x \to \infty} \frac{4 x^{2}}{6 - \frac{7}{x}}$

Step 6

As $x$ approaches $\infty$ , the fraction $\frac{7}{x}$ approaches $0$ .

$lim_{x \to \infty} \frac{4 x^{2}}{6 - 0}$

Step 7

Since its numerator is unbounded while its denominator approaches a constant number, the fraction $\frac{4 x^{2}}{6 - 0}$ approaches infinity.

$\infty$